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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2016/ib/paper-1/17c/b/solution

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Past exam of the mathematics course of the University of Cambridge / 2016 / ib / Paper 1 / 17C / b / Solution by

Codex  0 Created 2026-09-24 Updated 2026-10-06

r (s)

with

d r / d s = u (r (s))

. The chain rule gives

\frac{d}{d s} H (r (s)) = \nabla H \cdot \frac{d r}{d s} = u \cdot \nabla H = 0.

(1)

Therefore $H$ is constant along every streamline. A different regular parametrization merely multiplies this derivative by a scalar and gives the same conclusion. For time-dependent flow, the claim concerns streamlines at a fixed time, rather than Lagrangian trajectories.

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